Theorem conncompss 23462

Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypothesis

Ref	Expression
conncomp.2	⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}

Assertion

Ref	Expression
conncompss	⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑆)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋

Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem conncompss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑋)
2		conntop 23446	. . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Conn → (𝐽 ↾t 𝑇) ∈ Top)
3	2	3ad2ant3 1135	. . . . . 6 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝐽 ↾t 𝑇) ∈ Top)
4		restrcl 23186	. . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V))
5	4	simprd 495	. . . . . 6 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → 𝑇 ∈ V)
6		elpwg 4625	. . . . . 6 ⊢ (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
7	3, 5, 6	3syl 18	. . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
8	1, 7	mpbird 257	. . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ∈ 𝒫 𝑋)
9		3simpc 1150	. . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn))
10		eleq2 2833	. . . . . 6 ⊢ (𝑦 = 𝑇 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑇))
11		oveq2 7456	. . . . . . 7 ⊢ (𝑦 = 𝑇 → (𝐽 ↾t 𝑦) = (𝐽 ↾t 𝑇))
12	11	eleq1d 2829	. . . . . 6 ⊢ (𝑦 = 𝑇 → ((𝐽 ↾t 𝑦) ∈ Conn ↔ (𝐽 ↾t 𝑇) ∈ Conn))
13	10, 12	anbi12d 631	. . . . 5 ⊢ (𝑦 = 𝑇 → ((𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn) ↔ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn)))
14		eleq2 2833	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
15		oveq2 7456	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐽 ↾t 𝑥) = (𝐽 ↾t 𝑦))
16	15	eleq1d 2829	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐽 ↾t 𝑥) ∈ Conn ↔ (𝐽 ↾t 𝑦) ∈ Conn))
17	14, 16	anbi12d 631	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)))
18	17	cbvrabv 3454	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)}
19	13, 18	elrab2 3711	. . . 4 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn)))
20	8, 9, 19	sylanbrc 582	. . 3 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
21		elssuni 4961	. . 3 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
22	20, 21	syl 17	. 2 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
23		conncomp.2	. 2 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}
24	22, 23	sseqtrrdi 4060	1 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931  (class class class)co 7448   ↾_t crest 17480  Topctop 22920  Conncconn 23440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-rest 17482  df-top 22921  df-conn 23441

This theorem is referenced by:  conncompcld  23463  tgpconncompeqg  24141  tgpconncomp  24142